Duality of a great rhombiicosidodecahedron: (VRML)

The edges and vertices of a polyhedron constitute a special case of a graph, which is a set of N0 points or nodes, joined in pairs by N1 segments or branches. Hence, the essential property of a polyhedron is that its faces together form a single unbounded surface. The edges are merely curves drawn on the surface, which come together in sets of three or more at the vertices.

In other words, a polyhedron with N2 faces, N1 edges, and N0 vertices may be regarded as a map, i.e., as the partition of an unbounded surface into N2 polygonal regions by means of N1 simple curves joining pairs of N0 points.

From a given map, we may derive a second, called the dual map, on the same surface. This second map has N2 vertices, one in the interior of each face of the given map; N1 edges, one crossing each edge of the given map; and N0 faces, one surrounding each vertex of the given map. Corresponding to a p-gonal face of the given map, the dual map will have a vertex where p edges (and p faces) come together.

Duality is a symmetric relation: a map is the dual of its dual.

Regular map: a map is said to be regular, of type {p,q} if there are p vertices and p edges for each face, q edges and q faces at each vertex, arranged symmetrically in a sense that can be made precise. Thus, a regular polyhedron is a special case of a regular map. For each map of type {p,q} , there is a dual map of type {q,p} .

Consider the regular polyhedron {p,q} , with its N0 vertices, N1 edges, N2 faces. If we replace each edge by a perpendicular line touching the mid-sphere at the same point, we obtain the N1 edges of the reciprocal polyhedron {q,p} , which has N2 vertices and N0 faces. This process is in fact reciprocation with respect to the mid-sphere: the vertices and face-planes of {p,q} are the poles and polars of the face-planes and vertices of {q,p} .

Reciprocation with respect to another concentric sphere would yield a larger or smaller {q,p} .

This process of reciprocation can evidently be applied to any figure which has a recognizable "center". It agrees with the topological duality that we defined for maps. The thirteen Archimedean solids hence are included in this case; i.e., for each Archimedean solid, there exists a reciprocal polyhedron (with respect to a concentric sphere).

In Maple, one can define a duality of a regular polyhedron or Archimedean solid via the command duality(dualp,p,s); where dualp is the name of the reciprocal polyhedron of the given polyhedron p with respect to the sphere s which is concentric with p (i.e., s and p have the same center).

Tetrahedron and its dual VRML	Hexahedron and its dual VRML
Octahedron and its dual VRML	Dodecahedron and its dual VRML
Icosahedron and its dual VRML	Great Stellated Dodecahedron and its dual VRML
Great Dodecahedron and its dual VRML	Small Stellated Dodecahedron and its dual VRML

Great Icosahedron and its dual VRML

A given regular polyhedra is closed under duality, i.e., the duality of a regular polyhedron is also a regular polyhedron. It is not the case for the Archimedean solids, thougth. The following table shows the polyhedron type of the duals of the Archimedean solids:

To access information relating to the reciprocal of an Archimedean solid pgon , use the following function calls:

Truncated tetrahedron and its dual VRML	Truncated octahedron and its dual VRML
Truncated hexahedron and its dual VRML	Truncated icosahedron and its dual VRML
Truncated dodecahedron and its dual VRML	Cuboctahedron and its dual VRML
Icosidodecahedron and its dual VRML	Small rhombicuboctahedron and its dual VRML
Small rhombiicosidodecahedron and its dual VRML	Great rhombicuboctahedron and its dual VRML
Great rhombiicosidodecahedron and its dual VRML	Snub cube and its dual VRML

Snub dodecahedron and its dual VRML

The Thirteen Archimedean Solids and Their Reciprocals